= Solution
Generators define a surjection $R^k\to M$, and conversely images of the standard <basis> generate any quotient. Writing $\phi(m_i)=\sum_j a_{ij}m_j$ with $a_{ij}\in I$ and applying the adjugate to $XI-A$ gives a monic annihilating <polynomial> whose lower coefficients lie in $I$. Taking $\phi=1$ when $IM=M$ yields $xM=0$ with $x-1\in I$; for $M=\mathbb Z/3$ and $I=(2)$ take $x=3$. The Jacobson radical criterion follows by placing a nonunit $1-rs$ in a maximal <ideal>. It yields Nakayama’s lemma. Applying the <determinant> trick to preimages of a finite generating set constructs a <polynomial> right inverse to any surjective endomorphism, proving injectivity.
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